Abstract
Three-dimensional Gaussian functions have been shown useful in representing electron microscopy (EM) density maps for studying macromolecular structure and dynamics. Methods that require setting a desired number of Gaussian functions or a maximum number of iterations may result in suboptimal representations of the structure. An alternative is to set a desired error of approximation of the given EM map and then optimize the number of Gaussian functions to achieve this approximation error. In this article, we review different applications of such an approach that uses spherical Gaussian functions of fixed standard deviation, referred to as pseudoatoms. Some of these applications use EM-map normal mode analysis (NMA) with elastic network model (ENM) (applications such as predicting conformational changes of macromolecular complexes or exploring actual conformational changes by normal-mode-based analysis of experimental data) while some other do not use NMA (denoising of EM density maps). In applications based on NMA and ENM, the advantage of using pseudoatoms in EM-map coarse-grain models is that the ENM springs are easily assigned among neighboring grains thanks to their spherical shape and uniformed size. EM-map denoising based on the map coarse-graining was so far only shown using pseudoatoms as grains.
Highlights
Single-particle analysis is an electron microscopy (EM) technique that allows determining the structure at near-atomic resolutions for a large range of macromolecular complexes [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]
A function f(r) (r ∈ R3) can be approximated using Gaussian radial basis functions (RBFs) by fN(r) = ∑Ni=1 ωiKσ(‖r−ri‖), where Kσ(r) is the RBF kernel that is a Gaussian function with the standard deviation σ and the amplitude of 1, to which we refer as pseudoatom, N is the number of pseudoatoms, ri is the vector of the center coordinates of the ith pseudoatom, ‖r − ri‖ is the Euclidean distance between the vectors r and ri, and ωi > 0 is the weight of the ith pseudoatom
Δf is the effective range of values in the EM map, rj is the voxel location at which the given EM map is compared with its approximation, and V is the total number of evaluated voxels
Summary
Single-particle analysis is an electron microscopy (EM) technique that allows determining the structure at near-atomic resolutions for a large range of macromolecular complexes [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. A different approach is to parametrize a Gaussian Mixture Model (GMM) of the probability density function using expectation-maximization algorithm [41, 45] All these approaches require setting a desired (target) number of grains or a maximum number of BioMed Research International. We have found that symmetry is preserved in EM-map approximations with this strategy for typical values of the target approximation error such as 1–15% [30, 33, 42,43,44] This method uses spherical Gaussian functions of fixed standard deviation that we refer to as pseudoatoms. For algorithmic details (e.g., related to adding/removing grains), the reader is addressed to [43] that describes this method in detail
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